/*===========================================================================
 
 Copyright (C) 2002-2012 Yves Renard, Julien  Pommier.
 
 This file is a part of GETFEM++
 
 Getfem++  is  free software;  you  can  redistribute  it  and/or modify it
 under  the  terms  of the  GNU  Lesser General Public License as published
 by  the  Free Software Foundation;  either version 3 of the License,  or
 (at your option) any later version along with the GCC Runtime Library
 Exception either version 3.1 or (at your option) any later version.
 This program  is  distributed  in  the  hope  that it will be useful,  but
 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
 or  FITNESS  FOR  A PARTICULAR PURPOSE.  See the GNU Lesser General Public
 License and GCC Runtime Library Exception for more details.
 You  should  have received a copy of the GNU Lesser General Public License
 along  with  this program;  if not, write to the Free Software Foundation,
 Inc., 51 Franklin St, Fifth Floor, Boston, MA  02110-1301, USA.
 
===========================================================================*/

/**
   @file contact_continuation_load.cc
   @brief Nonlinear Problem with Friction (large strain).

   A rubber bar is submitted to a large torsion. If the standard
   solver (Newton) fails, one tries to find a more suitable initial
   approximation by numerical continuation with respect to the
   Dirichlet condition and then restart the standard solver.
   NOTE: The continuation is proposed only for the quasi-static case.
   
   This program is used to check that getfem++ is working. This is
   also a good example of use of Getfem++.
*/

#include "getfem/getfem_assembling.h" /* import assembly methods (and norms comp.) */
#include "getfem/getfem_export.h"   /* export functions (save solution in a file)  */
#include "getfem/getfem_regular_meshes.h"
#include "getfem/getfem_model_solvers.h"
#include "getfem/getfem_nonlinear_elasticity.h"
#include "getfem/getfem_Coulomb_friction.h"
#include "getfem/getfem_superlu.h"
#include "gmm/gmm.h"

using std::endl; using std::cout; using std::cerr;
using std::ends; using std::cin;

/* some Getfem++ types that we will be using */
using bgeot::base_small_vector; /* special class for small (dim<16) vectors */
using bgeot::base_node;  /* geometrical nodes(derived from base_small_vector)*/
using bgeot::base_vector;
using bgeot::scalar_type; /* = double */
using bgeot::size_type;   /* = unsigned long */
using bgeot::base_matrix; /* small dense matrix. */

/* definition of some matrix/vector types.  These ones are built
 * using the predefined types in Gmm++
 */
typedef getfem::modeling_standard_sparse_vector sparse_vector;
typedef getfem::modeling_standard_sparse_matrix sparse_matrix;
typedef getfem::modeling_standard_plain_vector  plain_vector;

/*
  structure for the frictional problem
*/
struct friction_problem {

  enum { DIRICHLET_BOUNDARY_NUM = 0, FRICTION_BOUNDARY_NUM = 1, NEUMANN_BOUNDARY_NUM = 2};
  getfem::mesh mesh;         /* the mesh */
  getfem::mesh_im mim;       /* the integration methods */
  getfem::mesh_fem mf_u;     /* main mesh_fem, for the elastostatic solution */
  getfem::mesh_fem mf_p;     /* mesh_fem for the pressure.                   */
  getfem::mesh_fem mf_rhs;   /* mesh_fem for the right hand side (f(x),..)   */
  getfem::mesh_fem mf_coef;  /* mesh_fem used to represent pde coefficients  */
  getfem::mesh_fem mf_vm;    /* mesh_fem used for the von Mises stress       */
  scalar_type p1, p2, p3;    /* elastic coefficients.                        */
  scalar_type LX, LY, LZ;    /* system dimensions                            */
  scalar_type lambda, mu;    /* Lame coefficients                            */
  scalar_type residual;      /* max residual for the iterative solvers       */
  bool is_dynamic;
  scalar_type rho;           /* density                                      */
  size_type nocontact_mass;
  std::string datafilename;
  bgeot::md_param PARAM;

  bool solve(plain_vector &U);
  void init(void);
  friction_problem(void) : mim(mesh), mf_u(mesh), mf_p(mesh), mf_rhs(mesh), mf_coef(mesh), mf_vm(mesh) {}
};


/* Read parameters from the .param file, build the mesh, set finite element
   and integration methods and selects the boundaries.

   (this is boilerplate code, not very interesting)
 */
void friction_problem::init(void) {
  std::string MESH_TYPE = PARAM.string_value("MESH_TYPE","Mesh type ");
  std::string FEM_TYPE  = PARAM.string_value("FEM_TYPE","FEM name");
  std::string FEM_TYPE_P  = PARAM.string_value("FEM_TYPE_P","FEM name for the pressure");
  std::string INTEGRATION = PARAM.string_value("INTEGRATION",
					       "Name of integration method");
  cout << "MESH_TYPE=" << MESH_TYPE << "\n";
  cout << "FEM_TYPE="  << FEM_TYPE << "\n";
  cout << "INTEGRATION=" << INTEGRATION << "\n";

  /* First step : build the mesh */
  bgeot::pgeometric_trans pgt = 
    bgeot::geometric_trans_descriptor(MESH_TYPE);
  size_type N = pgt->dim();
  
  LX = PARAM.real_value("LX", "Length along X axis");
  LY = PARAM.real_value("LY", "Length along Y axis");
  LZ = PARAM.real_value("LZ", "Length along Z axis");
  std::string meshfilename = PARAM.string_value("MESHFILENAME");
  int nb_refine = PARAM.int_value("NBREFINE") ? PARAM.int_value("NBREFINE") : 0;
  scalar_type layerx = PARAM.real_value("LAYERX",
					"Thickness of the refinement along the X-axis");
  scalar_type layery = PARAM.real_value("LAYERY",
					"Thickness of the refinement along the Y-axis");
  scalar_type layerx_fact = PARAM.real_value("LAYERX_FACT", "Multiplicative factor");
  scalar_type layery_fact = PARAM.real_value("LAYERY_FACT", "Multiplicative factor");
  lambda = PARAM.real_value("LAMBDA", "Lame coefficient lambda");
  mu = PARAM.real_value("MU", "Lame coefficient mu");

  if (meshfilename.size() > 0)
    mesh.read_from_file(meshfilename);
  else {
    std::vector<size_type> nsubdiv(N);
    std::fill(nsubdiv.begin(),nsubdiv.end(),
	      PARAM.int_value("NX", "Number of space steps "));
    nsubdiv[1] = PARAM.int_value("NY") ? PARAM.int_value("NY") : nsubdiv[0];
    if (N>2) nsubdiv[2] = PARAM.int_value("NZ") ? PARAM.int_value("NZ") : nsubdiv[0];
    getfem::regular_unit_mesh(mesh, nsubdiv, pgt,
			      PARAM.int_value("MESH_NOISED") != 0);

    bgeot::base_matrix M(N,N);
    for (size_type i=0; i < N; ++i) {
      static const char *t[] = {"LX","LY","LZ"};
      M(i,i) = (i<3) ? PARAM.real_value(t[i],t[i]) : 1.0;
    }
    if (N>1) { M(0,1) = PARAM.real_value("INCLINE") * PARAM.real_value("LY"); }
    
    /* scale the unit mesh to [LX,LY,..] and incline it */
    mesh.transformation(M);
  }

  if (nb_refine > 0) { /* refinement in the right-bottom corner */
    dal::bit_vector cvref;
    for (int i = 0; i < nb_refine; ++i) {
      cvref.clear();
      for (dal::bv_visitor j(mesh.convex_index()); !j.finished(); ++j) {
	if ((mesh.points_of_convex(j)[0][0] > LX * (1 - layerx)
	     && mesh.points_of_convex(j)[1][0] > LX * (1 - layerx)
	     && mesh.points_of_convex(j)[2][0] > LX * (1 - layerx))
	    && (mesh.points_of_convex(j)[0][N-1] < LY * layery
		&& mesh.points_of_convex(j)[1][N-1] < LY * layery
		&& mesh.points_of_convex(j)[2][N-1] < LY * layery))
	  cvref.add(j);
      }
      mesh.Bank_refine(cvref);
      layerx *= layerx_fact;
      layery *= layery_fact;
    }
  }

  datafilename = PARAM.string_value("ROOTFILENAME","Base name of data files.");
  residual = PARAM.real_value("RESIDUAL"); if (residual == 0.) residual = 1e-10;

  is_dynamic = (PARAM.int_value("DYNAMIC", "Is dynamic?") != 0);
  rho = PARAM.real_value("RHO", "Density");
  nocontact_mass = PARAM.int_value("NOCONTACT_MASS", "Suppress the mass "
				   "of contact nodes");

  p1 = PARAM.real_value("P1", "First Elastic coefficient");
  p2 = PARAM.real_value("P2", "Second Elastic coefficient");
  p3 = PARAM.real_value("P3", "Third Elastic coefficient");
  
  mf_u.set_qdim(bgeot::dim_type(N));

  /* set the finite element on the mf_u */
  getfem::pfem pf_u = 
    getfem::fem_descriptor(FEM_TYPE);
  getfem::pintegration_method ppi = 
    getfem::int_method_descriptor(INTEGRATION);

  mim.set_integration_method(ppi);
  mf_u.set_finite_element(pf_u);

  mf_p.set_finite_element(getfem::fem_descriptor(FEM_TYPE_P));

  /* set the finite element on mf_rhs (same as mf_u if DATA_FEM_TYPE is
     not used in the .param file */
  std::string data_fem_name = PARAM.string_value("DATA_FEM_TYPE");
  if (data_fem_name.size() == 0) {
    GMM_ASSERT1(pf_u->is_lagrange(), "You are using a non-lagrange FEM"
		". In that case you need to set "
		<< "DATA_FEM_TYPE in the .param file");
    mf_rhs.set_finite_element(mesh.convex_index(), pf_u);
  } else {
    mf_rhs.set_finite_element(mesh.convex_index(), 
			      getfem::fem_descriptor(data_fem_name));
  }
  
  /* set the finite element on mf_coef. Here we use a very simple element
   *  since the only function that need to be interpolated on the mesh_fem 
   * is f(x)=1 ... */
  mf_coef.set_finite_element(mesh.convex_index(),
			     getfem::classical_fem(pgt,0));

  mf_vm.set_classical_discontinuous_finite_element(1);

  /* set boundary conditions
   * (Dirichlet on the upper face, contact on the bottom face 
   *  and on parts of the adjacent faces, Neumann elsewhere) */
      cout << "Selecting the Dirichlet, Neumann and contact boundaries\n";
  getfem::mesh_region border_faces;
  getfem::outer_faces_of_mesh(mesh, border_faces);
  for (getfem::mr_visitor it(border_faces); !it.finished(); ++it) {
    assert(it.is_face());
    base_node un = mesh.normal_of_face_of_convex(it.cv(), it.f());
    un /= gmm::vect_norm2(un);
    if (gmm::abs(un[N-1] - 1.0) < 1.0E-7) { 
      mesh.region(DIRICHLET_BOUNDARY_NUM).add(it.cv(),it.f());
    } else if (gmm::abs(un[N-1] + 1.0) < 1.0E-7) {
      mesh.region(FRICTION_BOUNDARY_NUM).add(it.cv(),it.f());
    } else if (mesh.points_of_convex(it.cv())[0][N-1] < LY * 0.3)
      mesh.region(FRICTION_BOUNDARY_NUM).add(it.cv(),it.f());
  }
}

/* The Iteration object for corrections */
class iteration_corr {
  size_type maxiter; /* Max. number of iterations.                       */
  int noise;         /* if noise > 0 iterations are printed.             */
  double resmax;     /* maximum residual.                                */
  double diffmax;    /* maximum difference.                              */
  size_type nit;     /* iteration number.                                */
  double res;        /* last computed residual.                          */
  double diff;       /* last computed difference.                        */
public:
  void init(void) { 
    nit = 0; res = 0.0; diff = 0.0; 
    }

  iteration_corr(double r = 1.0E-8, double d = 1.0E-8, int noi = 0,
		 size_type mit = size_type(-1))
    : maxiter(mit), noise(noi), resmax(r), diffmax(d) { init(); }

  void  operator ++(int) {  nit++; }

  int get_noisy(void) const { return noise; }
  void set_res(double r) { res = r; }
  void set_diff(double d) { diff = d; }
  size_type get_iteration(void) const { return nit; }

  bool converged(void) { return (res <= resmax && diff <= diffmax); }

  bool finished(void) {
    if (noise > 0)
      cout << "iter " << nit << " residual " << gmm::abs(res)
	   << " difference " << gmm::abs(diff) << endl;
    return ((nit >= maxiter || res > 1.0E+200) || converged());
  }
};

/* The object for adaptation of the continuation to non-smoothness of
   the problem by means of active selection functions */
class selections {
  size_type row, nba;
  gmm::dense_matrix<bool> CH;
  gmm::dense_matrix<size_type> ACT;
  plain_vector LV;
public:
  void clear(void) { row = nba = 0; }
  selections(void) { clear(); }
  bool empty(void) { return (nba == 0); }

  template<typename CH_MATRIX, typename T_MATRIX, typename VECT>
  void proper_update
  (const VECT &contact_nodes, const CH_MATRIX &CH_, const T_MATRIX &TST, scalar_type limit,
   int noisy);

  template<typename MODEL_STATE, typename CH_MATRIX, typename VECT>
  bool compute_new_tangent
  (MODEL_STATE &MS, getfem::mdbrick_abstract<MODEL_STATE> &final_model,
   getfem::mdbrick_Dirichlet<MODEL_STATE> &DIRICHLET,
   getfem::mdbrick_Coulomb_friction<MODEL_STATE> &FRICTION, const VECT &contact_nodes,
   plain_vector grad_XI, const plain_vector &F2_start, const plain_vector &deltaF2,
   const plain_vector &X, plain_vector &T, CH_MATRIX &CH_, int noisy);

  template<typename MODEL_STATE>
  void determine_tangent_orientation
  (getfem::mdbrick_Coulomb_friction<MODEL_STATE> &FRICTION, const plain_vector &UN,
   const plain_vector &UT, const plain_vector &UT0, const plain_vector &LN,
   const plain_vector &LT, plain_vector &T);  
};

template<typename MAT>
void compute_condition_number(const MAT &M) {
/* computes the smallest and the largest eigenvalue of the matrix M
   and then its condition number */
  
  size_type N = gmm::mat_nrows(M);
  plain_vector V(N), U(N), W(N);
  plain_vector V2(N), U2(N);  
  gmm::fill_random(V);
  gmm::fill_random(V2);
  scalar_type lambda = 0., lambda2 = 0.;
  gmm::SuperLU_factor<scalar_type> SLU;
  SLU.build_with(M);
  
  for(size_type i = 0; i < 100; ++i) {
    gmm::scale(V, 1./gmm::vect_norm2(V));
    gmm::scale(V2, 1./gmm::vect_norm2(V2));
    SLU.solve(U, V);
    gmm::mult(M, V2, U2);
    
    lambda = gmm::vect_sp(U, V) / gmm::vect_sp(V, V);
    lambda2 = gmm::vect_sp(U2, V2) / gmm::vect_sp(V2, V2);
    
    gmm::add(V, gmm::scaled(U, -1./lambda), W);
//     if (i > 90)
//       cout << "iteration " << i << " norm(W) = " <<  gmm::vect_norm2(W)
// 	   << " lambda = " << lambda 
// 	   << " lambda2 = " << lambda2 << endl;
    if (gmm::vect_norm2(W) < 1E-2) break;
    
    gmm::copy(U, V);
    gmm::copy(U2, V2);
  }
  
  // cout << "Smallest eigenvalue : " << 1.0/lambda << endl;
  // cout << "Largest eigenvalue : " << lambda2 << endl;
  cout << "Condition number: " << lambda2 * lambda << endl;
  
}

template<typename MAT, typename VECT1, typename VECT2>
void compute_tangent(const MAT &A, const VECT1 &B, VECT2 &T, int noisy) {
/* computes a unit tangent at the point X to the curve given implicitly by the equation H = 0
   with the Jacobian grad H(X) = (A|B) which is positively oriented w.r.t. the vector T
   and saves it into T, i.e.:
   solves (A|B)T_+ = 0 & T.T_+ = 1;
   sets T = T_+ / ||T_+||; */

  bool singular_A = false;
  size_type N = gmm::mat_nrows(A);
  double rcond;
		
  try { /* whether A is singular; if not, the calculation can be accelerated provided that 
	   A is sparse  */
    scalar_type sp;
    plain_vector Y(N);
    if (noisy > 1) cout << "starting computing tangent" << endl;

    gmm::SuperLU_solve(A, Y, B, rcond);
    sp = gmm::vect_sp(gmm::sub_vector(T, gmm::sub_interval(0,N)), Y);
		  
    T[N] = 1.0 / (T[N] - sp);
    gmm::copy(gmm::scaled(Y, -1.0*T[N]),
	      gmm::sub_vector(T, gmm::sub_interval(0, N)));
  }
  catch (...) {
    singular_A = true;
  }
  
  if (singular_A) {
    sparse_matrix C(N+1, N+1);
    plain_vector R(N+1);

    gmm::copy(A, gmm::sub_matrix(C, gmm::sub_interval(0, N), gmm::sub_interval(0, N)));
    gmm::copy(gmm::col_vector(B),
	      gmm::sub_matrix(C, gmm::sub_interval(0, N), gmm::sub_interval(N, 1)));
    gmm::copy(gmm::row_vector(T),
	      gmm::sub_matrix(C, gmm::sub_interval(N, 1), gmm::sub_interval(0, N+1)));

    gmm::clear(R);
    R[N] = 1.0;		  
    
    gmm::SuperLU_solve(C, T, R, rcond);
  }

  gmm::scale(T, 1.0/gmm::vect_norm2(T));
}

template<typename MAT, typename VECT1, typename VECT2>
void correction_solver
(const MAT &A, const VECT1 &B, VECT2 &X, VECT2 &T, const VECT1 &P, iteration_corr &iter) {
/* computes one correction step of the Moore-Penrose method for continuing the implicit curve
   H = 0 with P = H(X), (A|B) = grad H(X), X and T being the approximations of the point
   and the tangent, respectively, i.e.:
   solves (A|B)T_+ = 0 & T.T_+ = 1;
   solves (A|B)dX = P; T.dX = 0;
   sets X = X - dX;
   sets T = T_+ / ||T_+||;  */

  bool singular_A = false;
  size_type N = gmm::mat_nrows(A);
  plain_vector dX(N+1);

  try { /* whether A is singular; if not, the calculation can be accelerated provided that 
	   A is sparse */
    scalar_type sp1, sp2;
    plain_vector Y1(N), Y2(N);

    compute_condition_number(A);

    gmm::SuperLU_factor<scalar_type> SLU;
    SLU.build_with(A);
    SLU.solve(Y1, B);
    SLU.solve(Y2, P);

    sp1 = gmm::vect_sp(gmm::sub_vector(T, gmm::sub_interval(0,N)), Y1);
    sp2 = gmm::vect_sp(gmm::sub_vector(T, gmm::sub_interval(0,N)), Y2);
		
    dX[N] = sp2 / (sp1 - T[N]);
    gmm::add(gmm::scaled(Y1, -1.0*dX[N]), Y2);
    gmm::copy(Y2, gmm::sub_vector(dX, gmm::sub_interval(0, N)));
		
    T[N] = 1.0 / (T[N] - sp1);
    gmm::copy(gmm::scaled(Y1, -1.0*T[N]),
	      gmm::sub_vector(T, gmm::sub_interval(0, N)) );
  }
  catch (...) {
    singular_A = true;
  }


  try {
    if (singular_A) {
      plain_vector Q(N+1), R(N+1);
      sparse_matrix C(N+1, N+1);

      gmm::copy(A, gmm::sub_matrix(C, gmm::sub_interval(0, N), gmm::sub_interval(0, N)));
      gmm::copy(gmm::col_vector(B),
		gmm::sub_matrix(C, gmm::sub_interval(0, N), gmm::sub_interval(N, 1)));
      gmm::copy(gmm::row_vector(T),
		gmm::sub_matrix(C, gmm::sub_interval(N, 1), gmm::sub_interval(0, N+1)));
		
      gmm::copy(P, gmm::sub_vector(Q, gmm::sub_interval(0, N)));
      Q[N] = 0.0;

      gmm::clear(R);
      R[N] = 1.0;	
		    
      gmm::SuperLU_factor<scalar_type> SLU;
      SLU.build_with(C);
      SLU.solve(dX, Q);
      SLU.solve(T, R);
    }
  }
  catch (...) {
    cout << "SuperLU failed!" << endl;
  }
    
  gmm::add(gmm::scaled(dX, -1.0), X);
  gmm::scale(T, 1.0/gmm::vect_norm2(T));

  iter.set_diff(gmm::vect_norm2(dX));

}

template<typename MODEL_STATE>
void standard_solver
(MODEL_STATE &MS, getfem::mdbrick_abstract<MODEL_STATE> &final_model,
 getfem::mdbrick_Dirichlet<MODEL_STATE> &DIRICHLET, const plain_vector &F2,
gmm::iteration &iter) {
  /* standard solver (Newton) */

  DIRICHLET.rhs().set(F2);
// cout << "F2 = " << F2 << endl;
// cout << "|U| = " << gmm::vect_norm2(MS.state()) << "\n";

  getfem::basic_newton_line_search ls(size_type(-1), 5.0/3.0, 0.1, 0.5, 3.0);
//   getfem::default_newton_line_search ls;
  getfem::standard_solve(MS, final_model, iter,
			 getfem::default_linear_solver(final_model), ls);
}

template<typename MODEL_STATE>
void Newton_correction
(MODEL_STATE &MS, getfem::mdbrick_abstract<MODEL_STATE> &final_model,
 getfem::mdbrick_Dirichlet<MODEL_STATE> &DIRICHLET, const plain_vector &grad_XI,
 const plain_vector &F2_start, const plain_vector &deltaF2, plain_vector &X, plain_vector &T,
 iteration_corr &iter) {

  size_type stot = gmm::mat_ncols(MS.tangent_matrix());
  plain_vector F2 = F2_start;

  gmm::copy(gmm::sub_vector(X, gmm::sub_interval(0, stot)), MS.state());
  gmm::add(gmm::scaled(deltaF2, X[stot]), F2);
  DIRICHLET.rhs().set(F2);

  final_model.compute_residual(MS);

  do {
    if (iter.get_noisy() > 1)
      cout << "starting computing tangent matrix" << endl;
    final_model.compute_tangent_matrix(MS);

    if (iter.get_noisy() > 1)
      cout << "starting linear solver" << endl;
    correction_solver(MS.tangent_matrix(), grad_XI, X, T, MS.residual(), iter);
    if (iter.get_noisy() > 1) 
      cout << "linear solver done" << endl;

    gmm::copy(gmm::sub_vector(X, gmm::sub_interval(0, stot)), MS.state());
    gmm::copy(F2_start, F2);
    gmm::add(gmm::scaled(deltaF2, X[stot]), F2);
    DIRICHLET.rhs().set(F2);

    final_model.compute_residual(MS);
    iter.set_res(gmm::vect_norm2(MS.residual()));

    iter++;
  } while (!iter.finished());
}

template<typename MODEL_STATE>
void compute_displacement
(MODEL_STATE &MS, getfem::mdbrick_abstract<MODEL_STATE> &final_model,
 getfem::abstract_hyperelastic_law &l,
 getfem::mdbrick_nonlinear_elasticity<MODEL_STATE> &ELAS_nonlin,
 getfem::mdbrick_isotropic_linearized_elasticity<MODEL_STATE> &ELAS_lin, plain_vector &U,
 size_type law_num) {

  if (law_num < 4) {
    l.reset_unvalid_flag();
    final_model.compute_residual(MS);
    if (l.get_unvalid_flag()) 
      GMM_WARNING1("The solution is not completely valid, the determinant "
		   "of the transformation is negative on "
		   << l.get_unvalid_flag() << " Gauss points");
    
    gmm::copy(ELAS_nonlin.get_solution(MS), U);
  } else
    gmm::copy(ELAS_lin.get_solution(MS), U);
}

template<typename MODEL_STATE>
void compute_von_Mises
(MODEL_STATE &MS, getfem::mdbrick_nonlinear_elasticity<MODEL_STATE> &ELAS_nonlin,
 getfem::mdbrick_isotropic_linearized_elasticity<MODEL_STATE> &ELAS_lin,
 getfem::mesh_fem &mf_vm, plain_vector &VM, size_type law_num) {
  /* computes the von Mises stress from the solution */
  if (law_num < 4)
    ELAS_nonlin.compute_Von_Mises_or_Tresca(MS, mf_vm, VM, false);
  else 
    ELAS_lin.compute_Von_Mises_or_Tresca(MS, mf_vm, VM, false);
}

template<typename MODEL_STATE>
scalar_type compute_test_function
(getfem::mdbrick_Coulomb_friction<MODEL_STATE> &FRICTION, const plain_vector &UN,
 const plain_vector &UT, const plain_vector &UT0, const plain_vector &LN,
 const plain_vector &LT, size_type i, size_type j) {
  
  scalar_type r = FRICTION.get_r(), alpha = FRICTION.get_alpha(), beta = FRICTION.get_beta();
  plain_vector gap(LN.size()), friction_coef(LN.size());
  scalar_type tst;
    
  gmm::copy(FRICTION.get_gap(), gap);
  gmm::copy(FRICTION.get_friction_coef(), friction_coef);

  switch (j) {
  case 0: tst = (LN[i] - r * alpha * (UN[i] - gap[i])) / r; break;
  case 1: tst = (-friction_coef[i] * LN[i] + LT[i] - r * beta * (UT[i] - UT0[i])) / r; break;
  default: tst = (friction_coef[i] * LN[i] + LT[i] - r * beta * (UT[i] - UT0[i])) / r; break;
  }

  return tst;
}

template<typename MODEL_STATE, typename T_MATRIX>
void compute_test_functions
(getfem::mdbrick_Coulomb_friction<MODEL_STATE> &FRICTION, const plain_vector &UN,
 const plain_vector &UT, const plain_vector &UT0, const plain_vector &LN,
 const plain_vector &LT, T_MATRIX &TST) {
  
  scalar_type r = FRICTION.get_r();
  scalar_type alpha = FRICTION.get_alpha(), beta = FRICTION.get_beta();
  plain_vector gap(LN.size()), friction_coef(LN.size());
    
  gmm::copy(FRICTION.get_gap(), gap);
  gmm::copy(FRICTION.get_friction_coef(), friction_coef);

  for (size_type i = 0; i < LN.size(); ++i) {
    TST(i, 0) = (LN[i] - r * alpha * (UN[i] - gap[i])) / r;
    TST(i, 1) = (- friction_coef[i] * LN[i] + LT[i] - r * beta * (UT[i] - UT0[i])) / r;
    TST(i, 2) = (friction_coef[i] * LN[i] + LT[i] - r * beta * (UT[i] - UT0[i])) / r;
  }
}

template<typename MODEL_STATE, typename CH_MATRIX, typename T_MATRIX, typename VECT>
void compute_activity
(getfem::mdbrick_Coulomb_friction<MODEL_STATE> &FRICTION, const VECT &contact_nodes,
 const plain_vector &UN, const plain_vector &UT, const plain_vector &UT0,
 const plain_vector &LN, const plain_vector &LT, CH_MATRIX &CH, T_MATRIX &TST, int noisy) {

  compute_test_functions(FRICTION, UN, UT, UT0, LN, LT, TST);

  if (noisy > 1)
    cout << "characters of the solution:" << endl;

  for (size_type i = 0; i < LN.size(); ++i) {
    CH(i, 0) = (TST(i, 0) <= 0);
    CH(i, 1) = (TST(i, 1) >= 0);
    CH(i, 2) = (TST(i, 2) <= 0);

    if (noisy > 1)
      cout << "node " << i << ": " << contact_nodes[i] << " "
	   << CH(i, 0) << CH(i, 1) << CH(i, 2)
//            << " " << TST(i, 0) << " " << TST(i, 1) << " " << TST(i, 2)
	   << endl;
  }
}

template<typename MODEL_STATE, typename CH_MATRIX, typename T_MATRIX, typename VECT>
size_type compute_activity
(getfem::mdbrick_Coulomb_friction<MODEL_STATE> &FRICTION, const VECT &contact_nodes,
 const plain_vector &UN, const plain_vector &UT, const plain_vector &UT0,
 const plain_vector &LN, const plain_vector &LT, const CH_MATRIX &CH0, CH_MATRIX &CH,
 T_MATRIX &TST, scalar_type x_min, int noisy) {
  
  size_type nch = 0, nch_i = 0;

  compute_activity(FRICTION, contact_nodes, UN, UT, UT0, LN, LT, CH, TST, noisy);

  for (size_type i = 0; i < LN.size(); ++i) {
    nch_i = 0;
    for (size_type j = 0; j < 3; ++j) {
      if ((CH(i, j) != CH0(i, j)) && (j == 0 || (j > 0 && (CH(i, 0) || CH0(i, 0))))
	  && contact_nodes[i][0] >= x_min)
	++nch_i;
    }

    if (nch_i > 0) {
      if (nch == 0)
	cout << "changes of characters:" << endl;
      cout << "node " << i << ": " << contact_nodes[i] << " "
	   << CH0(i, 0) << CH0(i, 1) << CH0(i, 2) << " -> "
	   << CH(i, 0) << CH(i, 1) << CH(i, 2) << endl;
      nch += nch_i;
    }
  }
  
  return nch;
}

template<typename MATRIX>
void straight_insertion
(MATRIX &M, plain_vector &KEY, size_type i) {
/* places the i-th row of M according to the absolut values of the corresponding components
   in KEY; increasing order is wanted in the first i components of KEY */

  bool found;
  size_type j = i;
  std::vector<size_type> X(2);
  scalar_type X_KEY;

  X[0] = M(i, 0); X[1] = M(i, 1);
  X_KEY = KEY[i];
  if (j == 0) found = true;
  else found = (gmm::abs(X_KEY) >= gmm::abs(KEY[j - 1]));
 
  while (!found) { // seeking the appropriate  position
    KEY[j] = KEY[j - 1];
    M(j, 0) = M(j - 1, 0);
    M(j, 1) = M(j - 1, 1);
    --j;
    if (j == 0) found = true;
    else found = (gmm::abs(X_KEY) >= gmm::abs(KEY[j - 1]));
  }

  M(j, 0) = X[0]; M(j, 1) = X[1];
  KEY[j] = X_KEY;
}

template<typename CH_MATRIX, typename T_MATRIX, typename VECT>
void selections::proper_update
(const VECT &contact_nodes, const CH_MATRIX &CH_, const T_MATRIX &TST, scalar_type limit,
 int noisy) {

  size_type nbc = gmm::mat_nrows(CH_);

  if (noisy > 1) {
    cout << "characters of the last computed solution:" << endl;
    for (size_type i = 0; i < nbc; ++i)
      cout << "node " << i << ": " << contact_nodes[i] << " " 
	   << CH_(i, 0) << CH_(i, 1) << CH_(i, 2) << endl;
  }

  if (gmm::mat_nrows(CH) == 0) {
    gmm::resize(CH, nbc, 3); gmm::resize(ACT, 3 * nbc, 2); gmm::resize(LV, 3 * nbc);
  }

  gmm::copy(CH_, CH);
  nba = 0;

  for (size_type i = 0; i < nbc; ++i) {   
    for (size_type j = 0; j < 3; ++j)
      if (gmm::abs(TST(i, j)) <= limit) {
	if (nba == 0)
	  cout << "test functions with values close to zero:" << endl;
	cout << "TST(" << i << ", " << j << ") = " << TST(i, j) << endl;
	
	ACT(nba, 0) = i; ACT(nba, 1) = j; LV[nba] = TST(i, j);
	straight_insertion(ACT, LV, nba);
	++nba;
      }
  }

  if (nba == 0)
    cout << "No test functions with values close to zero were founded. " << endl;
  if (nba >= nbc) {
    row = nba;
    cout << "The continuation path seems to be circular. " << endl;
  }
}

template<typename MODEL_STATE, typename CH_MATRIX, typename VECT>
bool selections::compute_new_tangent
(MODEL_STATE &MS, getfem::mdbrick_abstract<MODEL_STATE> &final_model,
 getfem::mdbrick_Dirichlet<MODEL_STATE> &DIRICHLET,
 getfem::mdbrick_Coulomb_friction<MODEL_STATE> &FRICTION, const VECT &contact_nodes,
 plain_vector grad_XI, const plain_vector &F2_start, const plain_vector &deltaF2,
 const plain_vector &X, plain_vector &T, CH_MATRIX &CH_, int noisy) {
  /* comes through the proposed Jacobians and changes exactly one character */

  bool new_tangent = false;
  size_type stot = gmm::mat_ncols(MS.tangent_matrix());
  plain_vector F2 = F2_start;
  
  if (row < nba) {
    gmm::copy(gmm::sub_vector(X, gmm::sub_interval(0, stot)), MS.state());
    gmm::add(gmm::scaled(deltaF2, X[stot]), F2);
    DIRICHLET.rhs().set(F2);
    
    size_type i, j;
    while (!new_tangent && row < nba) {
	
      i = ACT(row, 0), j = ACT(row, 1);
      CH(i, j) = !CH(i, j);
      
      if ((CH(i, 0) && (CH(i, 1) || CH(i, 2))) || (!CH(i, 0) && CH(i, 1) != CH(i, 2))) {
	/* the transition is meaningful */
	cout << "enforcing transition of node " << i << ": " << contact_nodes[i] << " ";
	switch (j) {
	case 0: cout << !CH(i, 0) << CH(i, 1) << CH(i, 2); break;
	case 1: cout << CH(i, 0) << !CH(i, 1) << CH(i, 2); break;
	case 2: cout << CH(i, 0) << CH(i, 1) << !CH(i, 2); break;
	}
	cout << " -> " << CH(i, 0) << CH(i, 1) << CH(i, 2) << endl;
	
	FRICTION.set_character_matrix(CH);
	final_model.compute_tangent_matrix(MS);
	
	compute_tangent(MS.tangent_matrix(), grad_XI, T, noisy);
	new_tangent = true; gmm::copy(CH, CH_);
      }
      
      CH(i, j) = !CH(i, j);
      ++row;
    }
    
    FRICTION.clear_character_matrix();
  }
  
  return new_tangent;
  
}

template<typename MODEL_STATE>
void selections::determine_tangent_orientation
(getfem::mdbrick_Coulomb_friction<MODEL_STATE> &FRICTION, const plain_vector &UN,
 const plain_vector &UT, const plain_vector &UT0, const plain_vector &LN,
 const plain_vector &LT, plain_vector &T) {

  size_type i = ACT(row - 1, 0), j = ACT(row - 1, 1);
  scalar_type tst = compute_test_function(FRICTION, UN, UT, UT0, LN, LT, i, j);

  if (LV[row - 1] * (tst - LV[row - 1])  > 0) {
    gmm::scale(T, -1.0);
    cout << "TST(i, j)(X0 + T) - TST(i, j)(X0) = " << LV[row - 1] - tst << ", ";
  } else
    cout << "TST(i, j)(X0 + T) - TST(i, j)(X0) = " << tst - LV[row - 1] << ", ";
}
  
/**************************************************************************/
/*  Model.                                                                */
/**************************************************************************/

bool friction_problem::solve(plain_vector &U) {
  
  size_type nb_dof_rhs = mf_rhs.nb_dof();
  size_type N = mesh.dim();
  size_type law_num = PARAM.int_value("LAW");
  // Elasticity brick.
  base_vector p(3); p[0] = p1; p[1] = p2; p[2] = p3;
  
  /* choose the material law */
  getfem::abstract_hyperelastic_law *pl = 0, *pCG = 0;
  switch (law_num) {
  case 0:
  case 1: pl = new getfem::SaintVenant_Kirchhoff_hyperelastic_law(); break;
  case 2:
    if (N < 3) {
      pCG = new getfem::Ciarlet_Geymonat_hyperelastic_law(); 
      pl = new getfem::plane_strain_hyperelastic_law(pCG);
    } else
      pl = new getfem::Ciarlet_Geymonat_hyperelastic_law();
    break;
  case 3: pl = new getfem::Mooney_Rivlin_hyperelastic_law(); break;
  case 4 : case 5 : break;
  default: GMM_ASSERT1(false, "no such law");
  }
  
  getfem::mdbrick_abstract<> *pELAS = 0;
  getfem::mdbrick_nonlinear_elasticity<> *pELAS_nonlin = 0;
  getfem::mdbrick_isotropic_linearized_elasticity<> *pELAS_lin = 0;
  
  if (law_num < 4) {
    p.resize(pl->nb_params());
    pELAS = pELAS_nonlin
      = new getfem::mdbrick_nonlinear_elasticity<>(*pl, mim, mf_u, p);
  } else {
    pELAS = pELAS_lin = new getfem::mdbrick_isotropic_linearized_elasticity<>
      (mim, mf_u, law_num == 5 ? 0.0 : lambda, mu);
  }
  
  getfem::mdbrick_abstract<> *pINCOMP = pELAS;
  switch (law_num) {
    case 1: 
    case 3: pINCOMP = new getfem::mdbrick_nonlinear_incomp<>(*pELAS, mf_p);
      break;
    case 5: {
      getfem::mdbrick_linear_incomp<> *pb = 
        new getfem::mdbrick_linear_incomp<>(*pELAS, mf_p);
      pINCOMP = pb;
      pb->penalization_coeff().set(1.0/lambda);
    }
  }

  size_type nb_step = int(PARAM.int_value("NBSTEP"));
  scalar_type deltat = PARAM.real_value("DELTAT", "Time step");
 
  // contact condition for the Lagrange elements
  dal::bit_vector cn = mf_u.basic_dof_on_region(FRICTION_BOUNDARY_NUM);
//   cout << "cn = " << cn << endl;
//   cout << "cn.card()/N = " << cn.card()/N << endl;
  sparse_matrix BN(cn.card()/N, mf_u.nb_dof());
  sparse_matrix BT((N-1)*cn.card()/N, mf_u.nb_dof());
  std::vector<base_node> contact_nodes;
  plain_vector gap(cn.card()/N);
  size_type jj = 0;
  for (dal::bv_visitor i(cn); !i.finished(); ++i)
    if (i % N == 0) {
//       cout << "point de contact " << mf_u.point_of_dof(i) << endl;
      BN(jj, i+N-1) = -1.;
      gap[jj] = mf_u.point_of_basic_dof(i)[N-1];
      contact_nodes.push_back(mf_u.point_of_basic_dof(i));
      for (size_type k = 0; k < N-1; ++k) BT((N-1)*jj+k, i+k) = 1.;
      ++jj;
    }
  
    // creating force density vectors
  int nbc = int(jj);
//   sparse_matrix MMBN(nbc, nbc), MMBT(nbc*(N-1), nbc*(N-1));
  plain_vector UN(nbc), UT(nbc*(N-1)), LN1(nbc), LT1(nbc*(N-1));
//   {
//     sparse_matrix BB(mf_u.nb_dof(), mf_u.nb_dof());
//     getfem::asm_mass_matrix(BB, mim, mf_u, mf_u, FRICTION_BOUNDARY_NUM);
//     std::vector<size_type> indN, indT;
//     for (dal::bv_visitor i(cn); !i.finished(); ++i)
//       if ((i%N) == N-1) indN.push_back(i); else indT.push_back(i);
//     gmm::sub_index SUBI(indN);
//     gmm::copy(gmm::sub_matrix(BB, SUBI, SUBI), MMBN);
//     gmm::sub_index SUBJ(indT);
//     gmm::copy(gmm::sub_matrix(BB, SUBJ, SUBJ), MMBT);    
//   }



  scalar_type friction_coef = PARAM.real_value("FRICTION_COEFF",
					       "Friction cefficient");
  scalar_type r = PARAM.real_value("R", "Augmentation parameter");
  scalar_type alpha = PARAM.real_value("ALPHA") ? PARAM.real_value("ALPHA") : 1.0;
  

  getfem::mdbrick_Coulomb_friction<> FRICTION(*pINCOMP, BN, gap,
					      friction_coef, BT);
  FRICTION.set_r(r);
  FRICTION.set_alpha(alpha);
  FRICTION.set_beta(1./deltat);
 
  // Defining the volumic source term.
  base_vector f(N);
  f[0] = PARAM.real_value("FORCEX","Amplitude of the gravity");
  f[1] = PARAM.real_value("FORCEY","Amplitude of the gravity");
  if (N>2)
    f[2] = PARAM.real_value("FORCEZ","Amplitude of the gravity");
  plain_vector F(nb_dof_rhs * N);
  for (size_type i = 0; i < nb_dof_rhs; ++i) {
    gmm::copy(f, gmm::sub_vector(F, gmm::sub_interval(i*N, N)));
  }

  getfem::mdbrick_source_term<> VOL_F(FRICTION, mf_rhs, F);

  // Dirichlet condition
  plain_vector F2(nb_dof_rhs * N);
  getfem::mdbrick_Dirichlet<> DIRICHLET(VOL_F, DIRICHLET_BOUNDARY_NUM);
  DIRICHLET.rhs().set(mf_rhs, F2);
  DIRICHLET.set_constraints_type(getfem::constraints_type
				   (PARAM.int_value("DIRICHLET_VERSION")));

  getfem::mdbrick_abstract<> *pfinal_model = &DIRICHLET;
  
  getfem::mdbrick_dynamic<> *pDYNAMIC = 0;

  // Eventual dynamic brick
  if (is_dynamic) {
    pfinal_model = pDYNAMIC = new getfem::mdbrick_dynamic<>(DIRICHLET, rho);
    pDYNAMIC->set_dynamic_coeff(1./(deltat*deltat), 1.);
    if (nocontact_mass)
      pDYNAMIC->no_mass_on_boundary(FRICTION_BOUNDARY_NUM, nocontact_mass == 2);
  }

  // Generic solver.
  getfem::standard_model_state MS(*pfinal_model);
  size_type step0 = PARAM.int_value("STEP0") ? PARAM.int_value("STEP0") : 0;
  bool start_standard_solver = (PARAM.int_value("STANDARD_SOLVER",
						"Start with the standard solver?") != 0);
  size_type step0_cont = start_standard_solver ? 0 : PARAM.int_value("STEP0_CONT");
  size_type range_cont = start_standard_solver ? 1 : PARAM.int_value("RANGE_CONT");
  size_type range_cont_inc = PARAM.int_value("RANGE_CONT_INC");
  std::string X0filename = start_standard_solver ? "" : PARAM.string_value("X0FILENAME");
  scalar_type h_init = PARAM.real_value("H_INIT", "Initial step length");
  scalar_type h = PARAM.real_value("H") ? PARAM.real_value("H") : h_init;
  scalar_type h_max = PARAM.real_value("H_MAX", "Maximal step length");
  scalar_type h_min = PARAM.real_value("H_MIN", "Minimal step length");
  scalar_type h_change = PARAM.real_value("H_CHANGE");
  scalar_type h_inc = PARAM.real_value("H_INC");
  scalar_type h_dec = PARAM.real_value("H_DEC");
  scalar_type maxdist = PARAM.real_value("DISTANCE");
  scalar_type XI_end = start_standard_solver ? 1. : PARAM.real_value("XI_END");
  size_type maxit = PARAM.int_value("MAXITER");
  gmm::iteration iter;

  scalar_type dy = PARAM.real_value("DIRICHLET_Y",
				    "Prescribed displacement in y");
  scalar_type dxv = PARAM.real_value("DIRICHLET_X_SPEED",
				     "Prescribed velocity in x");
  scalar_type x_min = PARAM.real_value("X_MIN");
  int noisy = int(PARAM.int_value("NOISY"));


  plain_vector U0 = U, UT0 = UT, V0(gmm::vect_size(U)), DF(gmm::vect_size(U));
  gmm::dense_matrix<bool> CH_(nbc, 3), CH0_(nbc, 3);
  gmm::dense_matrix<scalar_type> TST(nbc, 3);
  short convergence = -1;
//  plain_vector SumN(nb_step), SumT(nb_step),Pressure(nb_step);
//  plain_vector maxT(nb_step);
//  plain_vector maxN(nb_step);
//  plain_vector max(nb_step);
//  double Fmax =0;

  if (step0 > 0){ /* load the foregoing values */
    char s[100]; sprintf(s, "step%d", step0);
    gmm::vecload(datafilename + s + ".U", U0);
    gmm::vecload(datafilename + s + ".Y", MS.state());
    if (is_dynamic)
      gmm::vecload(datafilename + s + ".V", V0);
  }
 

  for (size_type step = step0; step < nb_step; ++step) {
    cout << "beginning of step " << step+1 
	     << ", number of variables : " << pfinal_model->nb_dof() << endl ;

    convergence = -1;

    do { /* seek the solution */
      
      if (is_dynamic) {
	gmm::mult(pDYNAMIC->get_M(), gmm::scaled(U0, 1./(deltat*deltat)), DF);
	gmm::mult_add(pDYNAMIC->get_M(), gmm::scaled(V0, 1./deltat), DF);
	pDYNAMIC->set_DF(DF);
      }
      FRICTION.set_WT(gmm::scaled(U0, -1.0));
      
      /* in the first eleven iterations, the body is compressed,
	 afterwards, it is pulling to the right while subject to a constant pressure */
      for (size_type i = 0; i < nb_dof_rhs; ++i) {
	F2[i*N+N-1] = (step < 10) ? (dy * step/10.0) : dy;
	//  F2[i*N+N-1] = dy;
	F2[i*N+N-2] = (step < 10) ? 0.0 : (step-10)*deltat*dxv;
	// F2[i*N+N-1] = dy+dy*3*step/nb_step;
      }

      if (start_standard_solver) { /* the standard solver */
	iter = gmm::iteration(residual, noisy, maxit ? maxit : 40000);
	standard_solver(MS, *pfinal_model, DIRICHLET, F2, iter);

	if (iter.converged()) {
	  convergence = 1;
	}
	else {
	  cout << "the standard solver has failed, ";
	  if (convergence == 0) /* the initial approximation was given by the continuation */
	    convergence = - 2;
	}
	  
      } // the standard solver

      if (convergence == -1) {
	/* the solution has not been found by the standard solver; 
	   proceed with continuation -- the continuation parameter XI is such that 
	   the actual Dirichlet condition (load) is F2_start + XI * (F2 - F2_start) */

	cout << "starting to continue" << endl;
	GMM_ASSERT1(PARAM.int_value("DIRICHLET_VERSION") == 0, "The continuation only "
		    "implemented for the Dirichlet condition with multipliers");
	GMM_ASSERT1(!is_dynamic,
		    "The continuation is proposed only for a quasi-static case");
	
	scalar_type difference = PARAM.real_value("DIFFERENCE");
	if (difference == 0.) difference = 1e-10;
	scalar_type minangle = PARAM.real_value("ANGLE");
	scalar_type limit = PARAM.real_value("LIMIT", "limit for the test functions");
	size_type maxit_corr = PARAM.int_value("MAXITER_CORR");
	size_type thr_corr = PARAM.int_value("THRESHOLD_CORR");
	size_type nb_step_cont = PARAM.int_value("NBSTEP_CONT");

	short new_point;
	scalar_type XI0;
	size_type stot = gmm::mat_ncols(MS.tangent_matrix());
	size_type ind =  DIRICHLET.first_ind();
	size_type sc = gmm::vect_size(DIRICHLET.get_CRHS());
	plain_vector UT_init(nbc*(N-1)), U_init(nb_dof_rhs * N), F2_start(nb_dof_rhs * N),
	  F2_end = F2, deltaF2(nb_dof_rhs * N), grad_XI(stot), X0(stot+1), X(stot+1),
	  T0(stot+1), T(stot+1), VM(mf_vm.nb_dof());;
	gmm::dense_matrix<bool> CH(nbc, 3), CH0(nbc, 3);
	gmm::dense_matrix<scalar_type> TST0(nbc, 3);

 	while (convergence == -1 && range_cont < step + 1) {
	  cout << "Continuation " << step - range_cont + 1 << " -> " << step + 1 << endl;

	  char s[100]; sprintf(s, "step%d", step - range_cont + 1);
	  gmm::vecload(datafilename + s + ".U", U_init);
	  FRICTION.set_WT(gmm::scaled(U_init, -1.0));
	  gmm::mult(BT, U_init, UT_init);

	  gmm::copy(F2_end, deltaF2);
	  gmm::vecload(datafilename + s + ".F2", F2_start);
	  gmm::add(gmm::scaled(F2_start, -1.0), deltaF2);
	  DIRICHLET.rhs().set(deltaF2);
	  pfinal_model->compute_tangent_matrix(MS);
	  gmm::copy(DIRICHLET.get_CRHS(),
		    gmm::sub_vector(grad_XI, gmm::sub_interval(ind, sc)));
	  gmm::scale(grad_XI, -1.0);
	  
	  new_point = -1;
	  if (step0_cont == 0 && (range_cont == 1 && X0filename.size() == 0)) {
	    /* the starting point together with the corresponding tangent is given by
	       the solution from the previous time step */
	    cout << "the starting point is given by the solution "
		 << "from the previous time step" << endl;
	    sprintf(s, "step%d", step);
	    gmm::vecload(datafilename + s + ".Y",  MS.state());
	    gmm::vecload(datafilename + s + ".U",  U);
	    gmm::vecload(datafilename + s + ".UN",  UN);
	    gmm::vecload(datafilename + s + ".UT",  UT);
	    gmm::vecload(datafilename + s + ".LN",  LN1);
	    gmm::vecload(datafilename + s + ".LT",  LT1);
	    gmm::vecload(datafilename + s + ".VM",  VM);
	    sprintf(s, "step%d", step - 1);
	    gmm::vecload(datafilename + s + ".UT",  UT0);
	    gmm::copy(MS.state(), gmm::sub_vector(X0, gmm::sub_interval(0, stot)));
	    XI0 = X0[stot] = 1. - 1. / range_cont;
	    
	    compute_activity(FRICTION, contact_nodes, UN, UT, UT0, LN1, LT1, CH0, TST0,
			     noisy);
	    // TST0 will not be up-to-date in the 1st iteration!!
	    DIRICHLET.rhs().set(F2_start);
	    FRICTION.set_character_matrix(CH0);
	    pfinal_model->compute_tangent_matrix(MS);
	    gmm::fill_random(T0);
	    compute_tangent(MS.tangent_matrix(), grad_XI, T0, noisy);
	    if (T0[stot] < 0) gmm::scale(T0, -1.0);
	    FRICTION.clear_character_matrix();

	    new_point = 1; 
	  } else {
	    if (step0_cont == 0) {
	      if (range_cont > 1) {
		/* an approximation of the starting point is given by
		   the solution from the previous time step */
		cout << "an approximation of the starting point is given by "
		     << "the solution from the previous time step" << endl;
		sprintf(s, "step%d", step);
		gmm::vecload(datafilename + s + ".Y",  MS.state());
		XI0 = X0[stot] = 1. - 1. / range_cont;
	      } else {
		/* an approximation of the starting point is loaded from a given file */
		cout << "an approximation of the starting point is loaded" << endl;
		gmm::vecload(X0filename, X0);
		gmm::copy(gmm::sub_vector(X0, gmm::sub_interval(0, stot)), MS.state());
		XI0 = X0[stot];
	      }

	      cout << "XI0 = " << XI0 << endl;	    
	      cout << "correcting the starting point" << endl;
	      gmm::copy(F2_start, F2);
	      gmm::add(gmm::scaled(deltaF2, XI0), F2);
	      iter = gmm::iteration(residual, noisy, maxit ? maxit : 40000);
	      standard_solver(MS, *pfinal_model, DIRICHLET, F2, iter);
	      
	      if (iter.converged()) {
		gmm::copy(MS.state(), gmm::sub_vector(X0, gmm::sub_interval(0, stot)));
		X0[stot] = XI0;
		
		/* the tangent is computed in the standard manner */
		pfinal_model->compute_tangent_matrix(MS); 
		gmm::fill_random(T0);
		compute_tangent(MS.tangent_matrix(), grad_XI, T0, noisy);
		if (T0[stot] < 0) gmm::scale(T0, -1.0);

		new_point = 1;
	      }
	    } else  {
	      /* the starting point is loaded together with the corresponding tangent */
	      cout << "the starting point is loaded" << endl;
	      char s1[100]; sprintf(s1, "step%d", step + 1);
	      char s2[100]; sprintf(s2, "_%d", range_cont);
	      char s3[100]; sprintf(s3, "_%d", step0_cont);
	      gmm::vecload(datafilename + s1 + s2 + s3 + ".X", X0);
	      gmm::copy(gmm::sub_vector(X0, gmm::sub_interval(0, stot)), MS.state());
	      XI0 = X0[stot];
	      gmm::vecload(datafilename + s1 + s2 + s3 + ".T", T0);
	      new_point = 1;
	    }
	    
	    if (new_point == 1) {
	      compute_displacement(MS, *pfinal_model, *pl, *pELAS_nonlin, *pELAS_lin, U,
				   law_num);
	      gmm::mult(BN, U, UN); gmm::mult(BT, U, UT);
	      gmm::copy(FRICTION.get_LN(MS), LN1); gmm::copy(FRICTION.get_LT(MS), LT1);
	      compute_activity(FRICTION, contact_nodes, UN, UT, UT_init, LN1, LT1, CH0, TST0,
			       noisy + 1);
	      compute_von_Mises(MS, *pELAS_nonlin, *pELAS_lin, mf_vm, VM, law_num);
	    }
	  }
	    
	  if (new_point == 1 && step0_cont == 0){
	    char s1[100]; sprintf(s1, "step%d", step + 1);
	    char s2[100]; sprintf(s2, "_%d", range_cont);
	    gmm::vecsave(datafilename + s1 + s2 + "_0.U", U);
	    gmm::vecsave(datafilename + s1 + s2 + "_0.UN", UN);
	    gmm::vecsave(datafilename + s1 + s2 + "_0.UT", UT);
	    gmm::vecsave(datafilename + s1 + s2 + "_0.LN", LN1);
	    gmm::vecsave(datafilename + s1 + s2 + "_0.LT", LT1);
	    gmm::vecsave(datafilename + s1 + s2 + "_0.X", X0);
	    gmm::vecsave(datafilename + s1 + s2 + "_0.VM", VM);
	  }
	  
	  size_type nb_dec;
	  selections sel;
	  iteration_corr iter_corr;
	  size_type step_cont = step0_cont;
	  h_init /= range_cont; h_max /= range_cont; h_min /= range_cont;
	  h_change /= range_cont; maxdist /= range_cont; 
	  
	  while (new_point == 1 && gmm::abs(XI_end - XI0) > maxdist && -XI0 < step
		 && step_cont < nb_step_cont) {
	    cout << "beginning of step " << step + 1 << "_" << range_cont << "_"
		 << step_cont + 1 << ", number of variables : " << stot + 1<< endl;
	    nb_dec = 0;
	    
	    do { /* seek a new point */
	      new_point = -1;
	      cout << "XI0 = " << XI0 << ", h = " << h <<  ", deltaXI = " << h * T0[stot]
		   << endl;
	      
	      gmm::copy(X0, X); gmm::copy(T0, T);
	      gmm::add(gmm::scaled(T, h), X);
	      
	      iter_corr = iteration_corr(residual, difference, noisy,
					 maxit_corr ? maxit_corr : 40000);
	      Newton_correction(MS, *pfinal_model, DIRICHLET, grad_XI, F2_start, deltaF2,
				X, T, iter_corr);
	      
	      if (iter_corr.converged()) {
		compute_displacement(MS, *pfinal_model, *pl, *pELAS_nonlin, *pELAS_lin, U,
				     law_num);
		gmm::mult(BN, U, UN); gmm::mult(BT, U, UT);
		gmm::copy(FRICTION.get_LN(MS), LN1); gmm::copy(FRICTION.get_LT(MS), LT1);
		size_type nb_change;
		if (step_cont == 0 && (range_cont == 1 && X0filename.size() == 0)) {
		  compute_activity(FRICTION, contact_nodes, UN, UT, UT_init, LN1, LT1, CH0,
				   CH, TST, x_min, noisy + 1);
		  nb_change = 0;
		} else
		  nb_change = compute_activity(FRICTION, contact_nodes, UN, UT, UT_init, LN1,
					       LT1, CH0, CH, TST, x_min, noisy - 1);
		
		scalar_type XI = X[stot], angle =  gmm::vect_sp(T0, T);
		cout << "XI = " << XI << ", XI - XI0 = " << XI - XI0 << ", T0.T = "
		     << angle;
		if (((nb_change == 0 && angle >= minangle)
		     || ((nb_change == 1 && h <= h_change) || step_cont == 0))
		    && XI <= XI_end + maxdist) {
// 		    && XI >= XI_end - maxdist) {
		  char s1[100]; sprintf(s1, "step%d", step + 1);
		  char s2[100]; sprintf(s2, "_%d", range_cont);
		  char s3[100]; sprintf(s3, "_%d", step_cont+1);
		  gmm::vecsave(datafilename + s1 + s2 + s3 + ".U", U);
		  gmm::vecsave(datafilename + s1 + s2 + s3 + ".UN", UN);
		  gmm::vecsave(datafilename + s1 + s2 + s3 + ".UT", UT);
		  gmm::vecsave(datafilename + s1 + s2 + s3 + ".LN", LN1);
		  gmm::vecsave(datafilename + s1 + s2 + s3 + ".LT", LT1);
		  gmm::vecsave(datafilename + s1 + s2 + s3 + ".X", X);
		  gmm::vecsave(datafilename + s1 + s2 + s3 + ".T", T);

		  compute_von_Mises(MS, *pELAS_nonlin, *pELAS_lin, mf_vm, VM, law_num);
		  gmm::vecsave(datafilename + s1 + s2 + s3 + ".VM", VM);
		  
		  XI0 = XI; gmm::copy(X, X0); gmm::copy(T, T0); gmm::copy(CH, CH0);
		  gmm::copy(TST, TST0);
		  if ((nb_dec == 0) && (iter_corr.get_iteration() <= thr_corr))
		    h = (h_inc * h < h_max) ? h_inc * h : h_max; 
		  sel.clear(); new_point = 1;
		  
		  cout << endl << "end of Step n° " << step + 1 << "_" << range_cont << "_"
		       << step_cont + 1 << endl;
		}
		else
		  cout << " - the point rejected" << endl;
	      }
	      
	      if (new_point <= 0) {
		if (h > h_min) {
		  h = (h_dec * h > h_min) ? h_dec * h : h_min;
		  ++nb_dec;
		  new_point = 0;
		} else { /* try to change the Jacobian */
		  if (sel.empty()) {
		    cout << "classical continuation has broken down, "
			 << "starting searching for a new Jacobian" << endl;
		    sel.proper_update(contact_nodes, CH0, TST0, limit, noisy);
		  }
		  gmm::copy(T0, T);
		  if (sel.compute_new_tangent(MS, *pfinal_model, DIRICHLET, FRICTION,
					      contact_nodes, grad_XI, F2_start, deltaF2,
					      X0, T, CH0, noisy)) {
		    gmm::add(gmm::sub_vector(X0, gmm::sub_interval(0, stot)),
			     gmm::sub_vector(T, gmm::sub_interval(0, stot)), MS.state());
		    compute_displacement(MS, *pfinal_model, *pl, *pELAS_nonlin, *pELAS_lin,
					 U, law_num);
		    gmm::mult(BN, U, UN); gmm::mult(BT, U, UT);
		    gmm::copy(FRICTION.get_LN(MS), LN1);
		    gmm::copy(FRICTION.get_LT(MS), LT1);
		    sel.determine_tangent_orientation(FRICTION, UN, UT, UT_init, LN1, LT1,
						      T);
		    cout << "T0.T = " << gmm::vect_sp(T0, T) << endl;
		    gmm::copy(T, T0);
		    
		    h = h_init; nb_dec = 0;
		    new_point = 0;
		  }
		}
	      }
	    } while (new_point == 0);
	      
	    ++step_cont;
	  } // the main loop of continuation
	  
	  if (new_point == 1 && gmm::abs(XI_end - XI0) <= maxdist) {
	    start_standard_solver = true; convergence = 0;
	    range_cont = 1; XI_end = 1.;
	    cout << "stop continuing, restarting the standard solver"
		 << " with a new initial approximation" << endl;
	  } else {
	    h = h_init = h_init * range_cont; h_max *= range_cont;
	    h_min *= range_cont; h_change *= range_cont; maxdist *= range_cont; 
	    range_cont += (range_cont == 1) ? 1 : range_cont_inc;
	    h /= range_cont;
	  }	  
	  step0_cont = 0; X0filename = "";

	} // loop of continuations for different values of range_cont
	
      } // continuation

    } while (convergence == 0);
    
    if (convergence == 1) { /* solution in the actual step has been found */

      compute_displacement(MS, *pfinal_model, *pl, *pELAS_nonlin, *pELAS_lin, U, law_num);
      gmm::mult(BN, U, UN); gmm::mult(BT, U, UT); gmm::mult(BT, U0, UT0);
      gmm::copy(FRICTION.get_LN(MS), LN1); gmm::copy(FRICTION.get_LT(MS), LT1);
      if (step == step0)
	compute_activity(FRICTION, contact_nodes, UN, UT, UT0, LN1, LT1, CH_, TST,
			 noisy + 1);
      else
	compute_activity(FRICTION, contact_nodes, UN, UT, UT0, LN1, LT1, CH0_, CH_, TST,
			 x_min, noisy);
      
      // gmm::copy(FRICTION.get_LN(MS), LN1);
      // gmm::copy(FRICTION.get_LT(MS), LT1);
      
      //     {
      //       gmm::iteration itercg(1e-12, 1);
      //       plain_vector LLN(nbc), LLT(nbc*(N-1));
      //       gmm::cg(MMBN, LLN, LN1, gmm::identity_matrix(), itercg);
      //       itercg.init();
      //       gmm::cg(MMBT, LLT, LT1, gmm::identity_matrix(), itercg);
      //     }
      /*    	cout << "LT1 = " << LT1 << endl;
	    cout << "LN1 = " << LN1 << endl;
      */
      
      
      //    plain_vector LT2(nbc);
      // 	maxT[step]=-LT1[0];
      // 	maxN[step]=-LN1[0];
      // Calculating forces on contact nodes and total contact pressure
      //    for (int y = 0; y < nbc; ++y) {
      //     
      //     if (N>2) 
      //         LT2[y] = sqrt(LT1[N*y]*LT1[N*y] + LT1[N*y +1]*LT1[N*y +1]);
      // 	
      // 	
      //     if (N<3) 
      //     	LT2[y]=LT1[y];
      // 	
      // // 	if (LT2[y]> maxT[step])
      // // 		maxT[step]=-LT2[y];
      // // 	
      // // 	if (LN1[y]> maxN[step])
      // // 		maxN[step]=-LN1[y];
      // 	
      // 	
      //     	SumT[step]+= -LT2[y];
      // 	SumN[step]+= -LN1[y]; 
      // 	Pressure[step]+= SumN[step]/(LX*LY);
      //     }
      //     cout << "Tu vas marcher BORDEL 3 \n";
      // // 	max[step]=sqrt(maxN[step]*maxN[step]+maxT[step]*maxT[step]);
      //    
      //     
      //     cout << "\n  \n \n normal contact pressure " << Pressure << endl;
      //     cout << "\n Total force along normal " << SumN << endl;
      //     cout << "\n Total force along tangential plane " << SumT << endl;
      // 
      // 	gmm::vecsave(datafilename + ".SumT",SumT);
      // 	gmm::vecsave(datafilename + ".SumN",SumN);
      // // 	cout << "LT2 = " << LT2 << endl;
      // 	cout << "LN2 = " << LN1 << endl;
      
      //    cout << "CN = " << FRICTION.get_LN(MS) << endl;

      //   plain_vector UN(gmm::mat_nrows(BN));
      //   gmm::mult(BN, U, UN);
      //    cout << "UN = " << UN << endl;
      
      char s[100]; sprintf(s, "step%d", step+1);
      gmm::vecsave(datafilename + s + ".F2", F2);
      gmm::vecsave(datafilename + s + ".U", U);
      gmm::vecsave(datafilename + s + ".Y", MS.state());
      gmm::vecsave(datafilename + s + ".UN", UN);
      gmm::vecsave(datafilename + s + ".UT", UT);
      gmm::vecsave(datafilename + s + ".LN", LN1);
      gmm::vecsave(datafilename + s + ".LT", LT1);
      
      
      plain_vector VM(mf_vm.nb_dof());
      compute_von_Mises(MS, *pELAS_nonlin, *pELAS_lin, mf_vm, VM, law_num);
      gmm::vecsave(datafilename + s + ".VM", VM);
      
      if (is_dynamic) {
	gmm::add(U, gmm::scaled(U0, -1.0), V0);
	gmm::scale(V0, 1./deltat);
	gmm::vecsave(datafilename + s + ".V", V0);
      }

      gmm::copy(U, U0); gmm::copy(CH_, CH0_);
      cout << "end of Step n° " << step+1 << " / " << nb_step << endl;
    
    // 	if (max[step]> Fmax)
    // 		Fmax = max[step];

    } else /* solution has not been found */
      break;
    
  } // the main cycle
  
//  cout << "end of all steps \n" ;
    if (law_num == 5 || law_num == 3 || law_num == 1) delete pINCOMP;
 /*  plain_vector VM(mf_u.nb_dof());
     cout << "calcul von mises\n";
   calcul_von_mises(mf_u, U0, mf_vm, VM, mu);
     cout << "Fin calcul von mises\n";
 */    

  return (convergence == 1);
  
}

  
/**************************************************************************/
/*  main program.                                                         */
/**************************************************************************/

int main(int argc, char *argv[]) {

  GMM_SET_EXCEPTION_DEBUG; // Exceptions make a memory fault, to debug.
  FE_ENABLE_EXCEPT;        // Enable floating point exception for Nan.

   // try {    
    friction_problem p;
    p.PARAM.read_command_line(argc, argv);
    p.init();
    p.mesh.write_to_file(p.datafilename + ".mesh");
    p.mf_u.write_to_file(p.datafilename + ".mf", true);
    p.mf_rhs.write_to_file(p.datafilename + ".mfd", true);
    p.mf_vm.write_to_file(p.datafilename + ".mfvm", true);
    plain_vector U(p.mf_u.nb_dof());
    if (!p.solve(U)) cerr << "Solve has failed\n";
    if (p.PARAM.int_value("VTK_EXPORT")) {
      cout << "export to " << p.datafilename + ".vtk" << "..\n";
      getfem::vtk_export exp(p.datafilename + ".vtk",
			     p.PARAM.int_value("VTK_EXPORT")==1);
      exp.exporting(p.mf_u); 
      exp.write_point_data(p.mf_u, U, "elastostatic_displacement");
      cout << "export done, you can view the data file with (for example)\n"
	"mayavi -d " << p.datafilename << ".vtk -f ExtractVectorNorm -f "
	"WarpVector -m BandedSurfaceMap -m Outline &\n";
    }
  // }   GMM_STANDARD_CATCH_ERROR;

  return 0;
  
}
